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INTEGRAL MEAN ESTIMATES FOR POLYNOMIALS
WHOSE ZEROS ARE WITHIN A CIRCLE
K. K. DEWAN, ABDULLAH MIR, and R. S. YADAV
(Received 3 November 2000)
Abstract.Letp(z)be a polynomial of degreenhaving all its zeros in|z| ≤k;k≤1, then for eachr >0,p >1,q >1 withp−1+q−1=1, Aziz and Ahemad (1996) recently proved thatn{2π
0 |p(eiθ)|rdθ}1/r≤ { 2π
0 |1+keiθ|prdθ}1/pr{ 2π
0 |p(eiθ)|qrdθ}1/qr. In this paper, we extend the above inequality to the class of polynomialsp(z)=anzn+ n
v=µan−vzn−v; 1≤µ≤nhaving all its zeros in|z| ≤k;k≤1 and obtain a generalization as well as a refinement of the above result.
2000 Mathematics Subject Classification. 30A10, 30C10, 30C15.
1. Introduction and statement of results. Letp(z)be a polynomial of degreen
and p(z) its derivative. If p(z)has all its zeros in |z| ≤1, then it was shown by Turan [7] that
max |z|=1
p(z)≥n 2max|z|=1
p(z). (1.1)
Inequality (1.1) is best possible with equality forp(z)=αzn+β, where|α| = |β|. As
an extension of (1.1) Malik [4] proved that ifp(z)has all its zeros in|z| ≤k, where
k≤1, then
max |z|=1
p(z)≥ n
1+kmax|z|=1
p(z). (1.2)
Malik [5] obtained a generalization of (1.1) in the sense that the right-hand side of (1.1) is replaced by a factor involving the integral mean of|p(z)|on|z| =1. In fact he proved the following theorem.
Theorem1.1. Ifp(z)has all its zeros in|z| ≤1, then for eachr >0
n 2π
0 p
eiθrdθ 1/r
≤
2π
0
1+eiθr dθ
1/r
max |z|=1
p(z). (1.3)
The result is sharp and equality in (1.3) holds forp(z)=(z+1)n.
Theorem1.2. Ifp(z)is a polynomial of degreenhaving all its zeros in|z| ≤k≤1, then forr >0,p >1,q >1with1/p+1/q=1,
n 2π
0 p
eiθrdθ 1/r
≤
2π
0
1+keiθqr dθ
1/qr2π
0
peiθpr dθ
1/pr . (1.4)
If we letr→ ∞andp→ ∞(so thatq→1) in (1.4) we get (1.2).
In this paper, we will first extendTheorem 1.2to the class of polynomialsp(z)=
anzn+n
v=µan−vzn−v, 1≤µ≤n, having all the zeros in|z| ≤k;k≤1, and thereby
obtain a generalization of it. More precisely, we prove the following result.
Theorem1.3. Ifp(z)=anzn+nv=µan−vzn−v,1≤µ≤nis a polynomial of de-green having all its zeros in|z| ≤k; k≤1, then for eachr >0, p >1, q >1with 1/p+1/q=1,
n 2π
0
peiθr dθ
1/r
≤
2π
0
1+kµeiθpr dθ
1/pr2π
0
peiθqr dθ
1/qr .
(1.5)
Remark1.4. If we letr→ ∞andq→ ∞(so thatp→1) in (1.5) we get (1.2) forµ=1.
Our next result is an improvement ofTheorem 1.3which in turn gives a generaliza-tion as well as a refinement ofTheorem 1.2.
Theorem1.5. Ifp(z)=anzn+nv=µan−vzn−v,1≤µ≤n, is a polynomial of degree
nhaving all its zeros in|z| ≤k;k≤1andm=min|z|=k|p(z)|, then for every real or complex numberβwith|β| ≤1,r >0,p >1,q >1with1/p+1/q=1,
n 2π
0
peiθ+βme i(n−1)θ kn−µ
rdθ 1/r ≤ 2π 0
1+kµeiθpr dθ
1/pr2π
0
peiθqr dθ
1/qr .
(1.6)
Remark1.6. Lettingr→ ∞and q→ ∞(so thatp→1) in (1.6) and choosing the argument ofβsuitably with|β| =1, it follows that, ifp(z)=anzn+n
v=µan−vzn−v,
1≤µ≤n, is a polynomial of degreenhaving all its zeros in|z| ≤k;k≤1, then
max |z|=1
p(z)≥1+nkµ
max |z|=1
p(z)+kn1−µmin|z|=1p(z)
. (1.7)
Inequality (1.7) was recently proved by Aziz and Shah [2].
2. Lemmas. For the proof ofTheorem 1.5we will make use of the following lem-mas.
Lemma2.1(see Aziz and Shah [2, Lemma 2]). Ifp(z)=anzn+nv=µan−vzn−vis a polynomial of degreenhaving all its zeros in|z| ≤k≤1, then
q(z)≤kµp(z) for|z| =1, 1≤µ≤n, (2.1)
Lemma2.2(see Rather [6]). Ifp(z)=anzn+nv=µan−vzn−v,1≤µ≤n, is a poly-nomial of degreenhaving all its zeros in|z| ≤k≤1, andm=min|z|=k|p(z)|, then
kµp(z)≥q(z)+ mn
kn−µ for|z| =1. (2.2)
3. Proof of theorems
Proof ofTheorem1.3. Suppose thatp(z)has all its zeros in|z| ≤k≤1, there-fore, byLemma 2.1we have
kµp(z)≥q(z) for|z| =1. (3.1)
Alsoq(z)=znp(1/z)¯ so thatp(z)=znq(1/z)¯, we have
p(z)=nzn−1q 1
¯
z
−zn−2q 1
¯
z
. (3.2)
Equivalently,
zp(z)=nznq 1
¯
z
−zn−1q 1
¯
z
(3.3)
which implies
p(z)=nq(z)−zq(z) for|z| =1. (3.4)
Using (3.1) in (3.4) we get
q(z)≤kµnq(z)−zq(z) for|z| =1; 1≤µ≤n. (3.5)
Sincep(z)has all its zeros in|z| ≤k≤1, by the Gauss-Lucas theorem all the zeros ofp(z)also lie in|z| ≤1. This implies that the polynomial
zn−1p 1 ¯
z
=nq(z)−zq(z) (3.6)
has all its zeros in|z| ≥1/k≥1.
Therefore, it follows from (3.5) that the function
w(z)= zq
(z)
kµnq(z)−zq(z) (3.7)
is analytic for |z| ≤1 and |w(z)| ≤1 for|z| ≤1. Furthermorew(0)=0.Thus the function 1+kµw(z) is subordinate to the function 1+kµz in |z| ≤1. Hence by a
well-known property of subordination [3] we have forr >0 and for 0≤θ <2π,
2π
0
1+kµweiθr dθ≤
2π
0
1+kµeiθr
Also from (3.7), we have
1+kµw(z)= nq(z)
nq(z)−zq(z) (3.9)
or
nq(z)=1+kµw(z)nq(z)−zq(z). (3.10)
Using (3.4) and also|p(z)| = |q(z)|in (3.10), we have
np(z)=1+kµw(z)p(z) for|z| =1. (3.11)
Combining (3.8) and (3.11) we get
nr 2π
0 p
eiθrdθ≤ 2π
0
1+kµeiθr
peiθrdθ forr >0. (3.12)
Now applying Hölder’s inequality forp >1,q >1 with 1/p+1/q=1 to (3.12), we get
nr 2π
0
peiθr dθ
≤2π
0
1+kµeiθpr dθ
1/p2π
0
peiθqr dθ
1/q
forr >0
(3.13)
which is equivalent to
n 2π
0
peiθr dθ
1/r
≤2π
0
1+kµeiθpr dθ
1/pr2π
0
peiθqr dθ
1/qr
forr >0
(3.14)
which proves the desired result.
Proof ofTheorem1.5. Sincep(z)has all its zeros in|z| ≤k≤1, therefore, by Lemma 2.2we get
kµp(z)≥q(z)+ mn
kn−µ for|z| =1,1≤µ≤n. (3.15)
Also by (3.4) for|z| =1, we have
p(z)=nq(z)−zq(z). (3.16)
Now using (3.15) for every complexβwith|β| ≤1, we get
q(z)+β¯mn kn−µ
≤q(z)+ mn kn−µ
≤kµp(z)
=kµnq(z)−zq(z) for|z| =1.
Sincep(z)has all its zeros in|z| ≤k≤1, the result follows on the same lines as that ofTheorem 1.3. Hence we omit the proof.
References
[1] A. Aziz and N. Ahemad,Integral mean estimates for polynomials whose zeros are within a circle, Glas. Mat. Ser. III31(51)(1996), no. 2, 229–237.MR 98d:30008. Zbl 874.30001. [2] A. Aziz and W. M. Shah,An integral mean estimate for polynomials, Indian J. Pure Appl.
Math.28(1997), no. 10, 1413–1419.MR 99a:30005. Zbl 895.30004.
[3] E. Hille,Analytic Function Theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, New York, 1962.MR 34#1490. Zbl 102.29401.
[4] M. A. Malik,On the derivative of a polynomial, J. London Math. Soc. (2)1(1969), 57–60. MR 40#2827. Zbl 179.37901.
[5] ,An integral mean estimate for polynomials, Proc. Amer. Math. Soc.91(1984), no. 2, 281–284.MR 85c:30007. Zbl 543.30002.
[6] N. A. Rather,Extremal properties and location of the zeros of polynomials, Ph.D. thesis, University of Kashmir, 1988.
[7] P. Turan,Über die Ableitung von Polynomen, Compositio Math.7(1939), 89–95 (German). MR 1,37b.
K. K. Dewan: Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi-110025, India
E-mail address:[email protected]
